General Mental Ability Alıştırma Soruları — State PSC Exam — Sayfa 4

Soru 61Soru →

Match each autonomous robotic surveyor (P, Q, R, S) executing a multi-step navigation sequence with its exact shortest straight-line displacement from its initial starting point.

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Cevap

The correct matching pairs are: Robotic Surveyor P matches with

13\text{ m}

, Robotic Surveyor Q matches with

15\text{ m}

, Robotic Surveyor R matches with

20\text{ m}

, and Robotic Surveyor S matches with

25\text{ m}

.

Each robotic surveyor's final position is determined by establishing a Cartesian coordinate system with the starting point at

(0,0)

. By breaking each movement into orthogonal components

(x, y)

using trigonometry for diagonal movements (

45^\circ, 135^\circ

) and applying the Pythagorean theorem

d = \sqrt{x^2 + y^2}

, we obtain exact straight-line displacements of

13\text{ m}

for P,

15\text{ m}

for Q,

20\text{ m}

for R, and

25\text{ m}

for S.

Adım Adım Çözüm

1

Calculate displacement coordinates for Robotic Surveyor P

Path vector components:
1.

15\text{ m}

North:

(0, 15)

2. Turn

135^\circ

CW (facing SE), walk

10\sqrt{2}\text{ m}

:

\Delta x = 10\sqrt{2}\sin(135^\circ) = 10

,

\Delta y = 10\sqrt{2}\cos(135^\circ) = -10 \implies (10, 5)

3. Turn

90^\circ

right (facing SW), walk

5\sqrt{2}\text{ m}

:

\Delta x = -5

,

\Delta y = -5 \implies (5, 0)

4. Walk

12\text{ m}

North:

\Delta x = 0

,

\Delta y = 12 \implies (5, 12)

Final distance:

d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = 13\text{ m}

.

Decomposing vectors into

x

(East) and

y

(North) components allows precise tracking of position.

2

Calculate displacement coordinates for Robotic Surveyor Q

Path vector components:
1.

20\text{ m}

East:

(20, 0)

2. Turn

135^\circ

CCW (facing NW), walk

8\sqrt{2}\text{ m}

:

\Delta x = -8

,

\Delta y = 8 \implies (12, 8)

3. Turn

45^\circ

CW (facing North), walk

7\text{ m}

:

\Delta x = 0

,

\Delta y = 7 \implies (12, 15)

4. Walk

6\text{ m}

South:

\Delta x = 0

,

\Delta y = -6 \implies (12, 9)

Final distance:

d = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = 15\text{ m}

.

Using trigonometry and coordinate tracking ensures correct evaluation of net displacement.

3

Calculate displacement coordinates for Robotic Surveyor R

Path vector components:
1.

10\text{ m}

West:

(-10, 0)

2. Turn

45^\circ

CCW (facing SW), walk

6\sqrt{2}\text{ m}

:

\Delta x = -6

,

\Delta y = -6 \implies (-16, -6)

3. Turn

135^\circ

CW from SW (facing North), walk

18\text{ m}

:

\Delta x = 0

,

\Delta y = 18 \implies (-16, 12)

Final distance:

d = \sqrt{(-16)^2 + 12^2} = \sqrt{256 + 144} = 20\text{ m}

.

Angular rotations must be measured relative to the current heading, not absolute cardinal directions.

4

Determine initial orientation and calculate displacement for Robotic Surveyor S

At sunrise, the Sun is in the East, so shadows fall toward the West. Facing its own shadow means S initially faces West.
1. Turn

90^\circ

right (facing North), walk

10\text{ m}

:

(0, 10)

2. Turn

45^\circ

right (facing NE), walk

7\sqrt{2}\text{ m}

:

\Delta x = 7

,

\Delta y = 7 \implies (7, 17)

3. Turn North, walk

7\text{ m}

:

\Delta x = 0

,

\Delta y = 7 \implies (7, 24)

Final distance:

d = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25\text{ m}

.

Shadow orientation establishes the initial cardinal direction from which relative turns are executed.

Anahtar Kavram

Vector Displacement, Angular Rotation, and Shadow Orientation

Soru 62Soru →

Read the following family scenario carefully and match each person in Column I with their exact relationship to $B$ listed in Column II.

Scenario:
In a family of six members— $A, B, C, D, E,$ and $F$ —spanning three generations:
- $D$ is married to $F$ , and $A$ is the father of $D$ .
- $C$ is the mother-in-law of $F$ and has only one child.
- $E$ is the daughter of $F$ , and $B$ is the sibling of $E$ .

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Cevap

The correct matches are:

A

matches with Paternal Grandfather,

C

matches with Paternal Grandmother,

D

matches with Father, and

E

matches with Sister.

By analyzing the three-generation structure:

A

(male) and

C

(female) are married parents of

D

(male).

D

is married to

F

(female). Their children are

E

(female) and

B

. Therefore, with respect to

B

:

A

is the paternal grandfather,

C

is the paternal grandmother,

D

is the father, and

E

is the sister.

Adım Adım Çözüm

1

Determine the parental generation relationships

A

is the father of

D

.

C

is the mother-in-law of

F

, who is married to

D

, which means

C

is the mother of

D

.

The mother-in-law of a spouse is the mother of the other spouse.

2

Identify the genders and marital tie of the middle generation

D

(male) and

F

(female) are married to each other.

D

is the only child of

A

and

C

.

A

is father and

C

is mother of

D

, making

D

their male child.

3

Determine the third generation relationships

E

(female) and

B

are children of

D

and

F

.

E

is the sister of

B

.

E

is the daughter of

F

and sibling of

B

.

4

Deduce each person's relationship to

B

A

is paternal grandfather,

C

is paternal grandmother,

D

is father, and

E

is sister.

Tracing lineage through

B

's father

D

yields paternal grandparents

A

and

C

, father

D

, and sister

E

.

Anahtar Kavram

Multi-generational blood relation tree deduction and kinship mapping

Tahmini Süre:1m 30s

Soru 63Soru →

An archaeological surveyor starting from a central benchmark walks $20\text{ m}$ due East. He then turns $135^\circ$ to his left and walks $14\sqrt{2}\text{ m}$ . Next, he turns $45^\circ$ to his left and walks $18\text{ m}$ . Finally, he turns $90^\circ$ to his right and walks $2\text{ m}$ to reach an excavation site. What is the shortest direct distance (in meters) between the starting benchmark and the excavation site?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The shortest direct distance between the starting benchmark and the excavation site is 20 meters.

By resolving each sequential movement into Cartesian coordinate components

(x, y)

, the final position relative to origin

(0,0)

is found to be

(-12\text{ m}, 16\text{ m})

. The straight-line distance is computed using the distance formula

\sqrt{(-12)^2 + (16)^2} = \sqrt{144 + 256} = \sqrt{400} = 20\text{ m}

.

Adım Adım Çözüm

1

Plot the initial displacement on a Cartesian coordinate plane with origin

(0,0)

at the starting benchmark.

Position after walking

20\text{ m}

East is

(20, 0)

facing East (

0^\circ

).

East represents the positive x-axis direction.

2

Calculate displacement after turning

135^\circ

left (facing North-West) and walking

14\sqrt{2}\text{ m}

.

Position becomes

(20 - 14, 0 + 14) = (6, 14)

.

In North-West direction, horizontal component is

-14\sqrt{2} \times \frac{1}{\sqrt{2}} = -14\text{ m}

and vertical component is

+14\sqrt{2} \times \frac{1}{\sqrt{2}} = +14\text{ m}

.

3

Calculate displacement after turning

45^\circ

left from North-West (facing West) and walking

18\text{ m}

.

Position becomes

(6 - 18, 14) = (-12, 14)

.

Turning

45^\circ

left from North-West aligns facing direction with West (negative x-axis).

4

Calculate displacement after turning

90^\circ

right from West (facing North) and walking

2\text{ m}

.

Final position is

(-12, 14 + 2) = (-12, 16)

.

Turning

90^\circ

right from West aligns facing direction with North (positive y-axis).

5

Calculate the direct straight-line distance from starting point

(0,0)

to final position

(-12, 16)

.

Distance

D = \sqrt{(-12)^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20\text{ m}

.

Applying the Pythagorean theorem

D = \sqrt{x^2 + y^2}

yields the shortest distance.

Anahtar Kavram

Coordinate Geometry and Vector Resolution in Direction and Distance Problems

Tahmini Süre:2m 0s

Soru 64Soru →

In a certain code language, the word GLIMMER is written as INKMOUT. Following the exact same logical transformation rule, what is the code for the word PROJECT?

RHQGPWV

RIQQGXV

RJQRGYV

JTKLUEF

Cevabı ve açıklamayı göster

Cevap: RHQGPWV

Cevap

The coded word for PROJECT is RHQGPWV.

The code follows a dual positional rule: letters at odd positions (1st, 3rd, 5th, 7th) are shifted forward by 2 positions in alphabetical order, while letters at even positions (2nd, 4th, 6th) are replaced by their reverse alphabetical counterpart (A↔Z, B↔Y, etc.) followed by a 1-letter backward shift. Applying these exact steps to PROJECT gives RHQGPWV.

Adım Adım Çözüm

1

Analyze the rule applied to letters at odd positions (1st, 3rd, 5th, 7th) in GLIMMER.

G (1st) → I (+2 steps forward), I (3rd) → K (+2 steps forward), M (5th) → O (+2 steps forward), R (7th) → T (+2 steps forward).

Establishes that letters at odd positions undergo a direct +2 forward alphabetical shift.

2

Analyze the rule applied to letters at even positions (2nd, 4th, 6th) in GLIMMER.

L (2nd, reverse letter O) → N (-1 step backward), M (4th, reverse letter N) → M (-1 step backward), E (6th, reverse letter V) → U (-1 step backward).

Establishes that letters at even positions are replaced by their opposite alphabetical letter (where A=Z, B=Y, etc.) and then shifted 1 letter backward.

3

Apply the odd-position transformation (+2 forward shift) to the target word PROJECT.

P (1st) → R, O (3rd) → Q, E (5th) → G, T (7th) → V.

Determines the output letters for positions 1, 3, 5, and 7.

4

Apply the even-position transformation (reverse letter - 1 step) to the target word PROJECT.

R (2nd, reverse I) → H, J (4th, reverse Q) → P, C (6th, reverse X) → W.

Determines the output letters for positions 2, 4, and 6.

5

Combine all transformed positions sequentially.

Position 1 to 7 yields RHQGPWV.

Finalizes the code for PROJECT.

Anahtar Kavram

Positional parity-based multi-step coding combining forward shift and reverse alphabetical indexing.

Soru 65Soru →

Which of the following terms logically completes the given alphanumeric series?

C24X, \quad F20U, \quad I16R, \quad L12O, \quad ?

O8L

N8L

O8M

O10L

Cevabı ve açıklamayı göster

Cevap: $O8L$

Cevap

O8L

The term containing

O8L

is correct because it satisfies all three independent series rules: the first letter advances by 3 (

L + 3 = O

), the middle number decreases by 4 (

12 - 4 = 8

), and the third letter shifts backward by 3 (

O - 3 = L

).

Adım Adım Çözüm

1

Analyze the pattern of the first letter in each term

The first letters follow the sequence:

C (3) \xrightarrow{+3} F (6) \xrightarrow{+3} I (9) \xrightarrow{+3} L (12) \xrightarrow{+3} O (15)

.

The first letter advances by 3 positions in alphabetical order.

2

Analyze the pattern of the numerical term

The numbers follow the sequence:

24 \xrightarrow{-4} 20 \xrightarrow{-4} 16 \xrightarrow{-4} 12 \xrightarrow{-4} 8

.

Each number decreases by a constant difference of 4.

3

Analyze the pattern of the last letter in each term

The last letters follow the sequence:

X (24) \xrightarrow{-3} U (21) \xrightarrow{-3} R (18) \xrightarrow{-3} O (15) \xrightarrow{-3} L (12)

.

The last letter moves backward by 3 positions in alphabetical order.

4

Combine the resulting components

Combining

O

,

8

, and

L

yields the missing term

O8L

.

Synthesizing all three independent rules completes the sequence.

Anahtar Kavram

Alphanumeric Series Completion with Simultaneous Forward-Reverse Letter Shifts and Arithmetic Decrement

Tahmini Süre:1m 0s

Soru 66Soru →

In a single row of students facing North, Akash is ranked $18^{\text{th}}$ from the left end, while Banya is ranked $24^{\text{th}}$ from the right end. Chetan is sitting $6^{\text{th}}$ to the right of Akash and $4^{\text{th}}$ to the left of Banya. If Akash and Banya interchange their positions while Chetan remains in his place, and then Deepa sits exactly midway between Akash's new position and Chetan's position, what is Deepa's rank from the left end of the row?

26^{\text{th}}

21^{\text{st}}

25^{\text{th}}

27^{\text{th}}

Cevabı ve açıklamayı göster

Cevap: $26^{\text{th}}$

Cevap

The rank of Deepa from the left end of the row is

26^{\text{th}}

.

By following the relative linear placements, Chetan's left rank is

18 + 6 = 24

, and Banya's left rank is

24 + 4 = 28

. After Akash swaps positions with Banya, Akash's new position from the left becomes

28

. Deepa sits directly midway between Chetan (

24^{\text{th}}

) and Akash's new position (

28^{\text{th}}

), which corresponds to the rank

\frac{24 + 28}{2} = 26^{\text{th}}

.

Adım Adım Çözüm

1

Determine Chetan's position from the left end.

Since Akash is

18^{\text{th}}

from the left and Chetan is

6^{\text{th}}

to his right, Chetan's left-rank

= 18 + 6 = 24^{\text{th}}

.

Moving right along a row facing North increases the positional index from the left.

2

Determine Banya's position from the left end.

Chetan is

4^{\text{th}}

to the left of Banya, so Banya's left-rank

= 24 + 4 = 28^{\text{th}}

.

Since Chetan is to the left of Banya, adding

4

to Chetan's position gives Banya's position from the left.

3

Find Akash's position after interchanging with Banya.

Akash takes Banya's original place, making Akash's new left-rank

= 28^{\text{th}}

.

Interchanging positions swaps their exact ranks from the left end.

4

Calculate Deepa's position midway between Chetan's position (

24^{\text{th}}

) and Akash's new position (

28^{\text{th}}

).

Midpoint

= \frac{24 + 28}{2} = 26^{\text{th}}

from the left end.

The exact midway index between two positions

p_1

and

p_2

is their arithmetic mean

(p_1 + p_2) / 2

.

Anahtar Kavram

Order and Ranking involving multi-person relative placement and position interchange

Tahmini Süre:2m 0s

Soru 67Soru →

Analyze the pattern in the numerical sequence given below:

4, \quad 9, \quad 20, \quad 43, \quad 90, \quad ?

What is the value of the missing term that completes the sequence?

Cevabı ve açıklamayı göster

Cevap: 185

Cevap

185

Each consecutive term is calculated by doubling the previous term and adding an increasing integer (

+1, +2, +3, +4, +5

). Following this rule,

90 \times 2 + 5 = 185

.

Adım Adım Çözüm

1

Examine differences or operations between adjacent terms

9 - 4 = 5

,

20 - 9 = 11

,

43 - 20 = 23

,

90 - 43 = 47

Identify the primary numerical increment pattern

2

Analyze the rule for term progression

The operations follow

T_n = T_{n-1} \times 2 + (n-1)

for

n \ge 2

Confirm consistency across all given terms

3

Compute the sixth term (

T_6

)

T_6 = 90 \times 2 + 5 = 185

Apply the addition increment

+5

following

+4

Anahtar Kavram

Number and Alphabet Series

Soru 68Soru →

Eight architects—Harper, Ingram, Jenkins, Knox, Larson, Miller, Novak, and Ortiz—are sitting in two parallel rows containing four people each, such that there is an equal distance between adjacent persons.

In Row 1, Harper, Ingram, Jenkins, and Knox are seated, and all of them are facing South.
In Row 2, Larson, Miller, Novak, and Ortiz are seated, and all of them are facing North.
Therefore, each person in Row 1 faces a person in Row 2.

The following information is given about their seating arrangement:
- Jenkins does not sit at an extreme end of the row.
- Harper sits second to the left of Jenkins.
- Larson faces Harper and sits to the immediate right of Ortiz.
- Novak sits at one of the extreme ends of the row but does not face Jenkins.
- Only one person sits between Miller and Larson.
- Ingram faces Novak.

Based on the given arrangement, who sits to the immediate left of Knox?

Harper

Jenkins

Ortiz

Miller

Cevabı ve açıklamayı göster

Cevap: Harper

Cevap

Harper

Based on the seating clues, Row 1 (South-facing) from viewer's left to right is: Ingram, Jenkins, Knox, and Harper. Knox is seated in the third position. Because he is facing South, his 'left' points toward the viewer's right. Therefore, the person sitting to his immediate left is Harper, who is in the fourth position.

Adım Adım Çözüm

1

Establish the directional framework for both rows.

Row 1 faces South (their left is our right). Row 2 faces North (their left is our left). Let's number positions 1 to 4 from our left to right.

Correctly orienting left and right based on facing direction is the most critical step to avoid inversion errors.

2

Place Jenkins and Harper in Row 1 using the first two clues.

Jenkins is at position 2 and Harper is at position 4.

Jenkins cannot be at extreme ends (1 or 4). Harper is second to the left of Jenkins. Since Row 1 faces South, 'left' means moving to our right. The only way Jenkins can have someone two spots to their left without being on an edge is if Jenkins is at position 2 and Harper is at position 4.

3

Place Larson and Ortiz in Row 2.

Larson is at position 4 and Ortiz is at position 3.

Larson faces Harper (position 4). Larson sits to the immediate right of Ortiz. Since Row 2 faces North, 'right' is our right, meaning Ortiz must be just left of Larson at position 3.

4

Place Miller and Novak in Row 2.

Miller is at position 2 and Novak is at position 1.

There is one person between Miller and Larson (position 4), forcing Miller to position 2. Novak is at an extreme end, leaving only position 1 open. Novak at position 1 does not face Jenkins (at position 2), satisfying the condition.

5

Complete Row 1 by placing Ingram and Knox.

Ingram is at position 1 and Knox is at position 3.

Ingram faces Novak (position 1). This leaves only position 3 for Knox.

6

Determine who sits to the immediate left of Knox.

Harper sits to the immediate left of Knox.

Knox is at position 3 facing South. His immediate left is our right, which is position 4. Harper occupies position 4.

Anahtar Kavram

Linear Seating Arrangement with Multiple Facing Directions

Soru 69Soru →

In a regional agricultural department, a survey was conducted among $160$ field officers regarding their supervisory duties across three major crop initiatives: Wheat ( $W$ ), Rice ( $R$ ), and Cotton ( $C$ ). The survey revealed that $85$ officers supervise Wheat, $70$ supervise Rice, and $60$ supervise Cotton. Furthermore, $35$ officers supervise both Wheat and Rice, $25$ supervise both Rice and Cotton, $30$ supervise both Wheat and Cotton, and $15$ officers supervise all three crop initiatives. How many field officers supervise exactly two of these three crop initiatives?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The total number of field officers who supervise exactly two crop initiatives is 45.

To find the number of officers supervising exactly two crop initiatives, we calculate the number of officers in each pair of overlapping sets exclusive of the three-set intersection: for Wheat and Rice only (

35 - 15 = 20

), for Rice and Cotton only (

25 - 15 = 10

), and for Wheat and Cotton only (

30 - 15 = 15

). Summing these three regions gives

20 + 10 + 15 = 45

.

Adım Adım Çözüm

1

Extract intersection values for each pair of sets and subtract the central intersection of all three sets

Only Wheat and Rice =

35 - 15 = 20

; Only Rice and Cotton =

25 - 15 = 10

; Only Wheat and Cotton =

30 - 15 = 15

The given pairwise intersections include officers who supervise all three initiatives. To isolate those who supervise exactly two, the three-set intersection must be removed from each pairwise intersection.

2

Sum the three distinct 'exactly two' regions

Total =

20 + 10 + 15 = 45

Adding these disjoint counts yields the total number of officers supervising precisely two crop initiatives.

Anahtar Kavram

Inclusion-Exclusion Principle and Venn Diagram Region Analysis for Three Overlapping Sets

Soru 70Soru →

In a family of six members— $P, Q, R, S, T,$ and $U$ —there are two married couples. $Q$ is a doctor and the mother of $T$ . $U$ is the grandfather of $R$ and works as a contractor. $S$ is the father of $T$ and works as a manager. $P$ is the sister of $R$ . How is $T$ related to $U$ ?

Grandson

Granddaughter

Cannot be determined

Nephew

Cevabı ve açıklamayı göster

Cevap: Cannot be determined

Cevap

Cannot be determined

The correct answer is 'Cannot be determined' because

U

is established as the grandfather of

T

, but the stem provides no gender information for

T

. Therefore,

T

could be either a grandson or a granddaughter of

U

.

Adım Adım Çözüm

1

Determine the parents of

T

Q

is the mother of

T

and

S

is the father of

T

.

This establishes that

Q

and

S

form one of the married couples.

2

Link

P, R,

and

T

into the family generation

P

is the sister of

R

, and

P, R, T

are siblings.

U

is the grandfather of

R

.

Since

U

is the grandfather of

R

,

U

is also the grandfather of

R

's siblings, including

T

.

3

Evaluate the gender of

T

The text defines

Q

as female,

S

as male,

U

as male, and

P

as female (sister), but provides no gender indicator for

T

.

In blood relation problems, gender cannot be assumed. Thus,

T

could be either a grandson or a granddaughter.

Anahtar Kavram

Gender Ambiguity in Blood Relations

Tahmini Süre:1m 15s

Soru 71Soru →

In a class merit list, Rahul is ranked $15^{\text{th}}$ from the top and $21^{\text{st}}$ from the bottom. What is the total number of students in the class?

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

The total number of students in the class is

35

.

When the position of a single person is given from both ends of a line, the total count is calculated using the formula

\text{Total} = \text{Position from Left/Top} + \text{Position from Right/Bottom} - 1

. Substituting the values gives

15 + 21 - 1 = 35

.

Adım Adım Çözüm

1

Identify the given ranks from both ends for the same person.

Rank from top = 15, Rank from bottom = 21.

The position of Rahul is provided relative to both the start and end of the queue.

2

Apply the single-person dual rank total calculation formula.

\text{Total} = \text{Rank}_{\text{top}} + \text{Rank}_{\text{bottom}} - 1

Since Rahul is included in both rank counts, adding the ranks counts him twice, requiring a subtraction of 1.

3

Perform the computation.

15 + 21 - 1 = 35

Simplifying the arithmetic yields the exact total headcount of the class.

Anahtar Kavram

Single-Person Dual Rank Total Calculation

Soru 72Soru →

In a district administrative audit of $240$ agricultural cooperatives, compliance with three digital platforms was surveyed: E-Farming Portal ( $P$ ), Soil Health Database ( $S$ ), and Crop Insurance Registry ( $I$ ). It was found that $115$ cooperatives use $P$ , $110$ use $S$ , and $105$ use $I$ . Exactly $25$ cooperatives do not use any of the three platforms, and $20$ cooperatives use all three platforms. Furthermore, the number of cooperatives using only $P$ , only $S$ , and only $I$ are in the ratio $8 : 9 : 7$ , respectively. How many cooperatives use exactly two of these digital platforms?

75

95

55

90

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

The number of cooperatives using exactly two digital platforms is 75.

By applying the principle of inclusion-exclusion across the 3 sets, the union of all three platforms is

240 - 25 = 215

. Expressing the total elements as the sum of single-platform users

(a+b+c)

, exactly-two-platform users

(d+e+f)

, and all-three-platform users (

20

), we get

(a+b+c) + (d+e+f) = 195

. Summing the individual set totals without the central intersection yields

(a+b+c) + 2(d+e+f) = 95 + 90 + 85 = 270

. Subtracting the first equation from the second eliminates

(a+b+c)

directly, leaving

d+e+f = 270 - 195 = 75

.

Adım Adım Çözüm

1

Calculate the total number of cooperatives using at least one digital platform.

Total using at least one platform =

240 - 25 = 215

.

Subtracting the cooperatives that use none of the platforms from the total audited population gives the union of all three sets

|P \cup S \cup I|

.

2

Set up equations for single-region and double-region intersection counts using set partition variables.

Let

a, b, c

be the counts for only

P

, only

S

, and only

I

respectively. Let

d, e, f

be the counts for exactly two platforms (

P \cap S

only,

S \cap I

only,

P \cap I

only). The count for all three platforms is

g = 20

.

Dividing the 3-set Venn diagram into 7 mutually exclusive regions allows exact algebraic formulation.

3

Express total set union and individual set totals in terms of these regions.

Equation 1 (Union):

(a + b + c) + (d + e + f) + 20 = 215 \implies (a + b + c) + (d + e + f) = 195

.
Individual sets:

|P| = a + d + f + 20 = 115 \implies a + d + f = 95

|S| = b + d + e + 20 = 110 \implies b + d + e = 90

|I| = c + e + f + 20 = 105 \implies c + e + f = 85

.

Each individual set sum accounts for its unique region, adjacent two-set intersections, and the three-set intersection.

4

Sum the three individual set equations and solve for

(d + e + f)

.

Equation 2 (Sum of set equations):

(a + b + c) + 2(d + e + f) = 95 + 90 + 85 = 270

.
Subtracting Equation 1 from Equation 2:

[(a + b + c) + 2(d + e + f)] - [(a + b + c) + (d + e + f)] = 270 - 195

(d + e + f) = 75

.

Subtracting the union sum eliminates the single-region terms

(a + b + c)

directly, isolating the sum of regions representing exactly two platforms.

Anahtar Kavram

Principle of Inclusion-Exclusion and Region Partitioning in 3-Set Venn Diagrams

Tahmini Süre:3m 0s

Soru 73Soru →

At sunset, a surveyor facing the setting sun launches a drone from a fixed control pad. The drone executes the following sequence of movements:
1. It travels $40\text{ m}$ straight ahead in the direction the surveyor is facing.
2. It turns $135^\circ$ clockwise and flies $30\sqrt{2}\text{ m}$ .
3. It makes a $90^\circ$ right turn and flies $10\sqrt{2}\text{ m}$ .
4. Finally, it turns $135^\circ$ counter-clockwise and flies $15\text{ m}$ .

What is the shortest straight-line distance of the drone from the control pad, and in which direction is it located relative to the starting point?

35\text{ m}

, North

35\text{ m}

, South-West

25\text{ m}

, North-West

95\text{ m}

, North

Cevabı ve açıklamayı göster

Cevap: $35\text{ m}$ , North

Cevap

The drone is

35\text{ m}

away from the control pad in the North direction.

At sunset, facing the setting sun means facing West. The drone travels

40\text{ m}

West (

x = -40, y = 0

). Turning

135^\circ

clockwise reorients it to North-East; flying

30\sqrt{2}\text{ m}

changes position to

x = -10, y = 30

. Turning

90^\circ

right reorients it to South-East; flying

10\sqrt{2}\text{ m}

shifts position to

x = 0, y = 20

. Turning

135^\circ

counter-clockwise reorients it North; flying

15\text{ m}

brings it to

x = 0, y = 35

. The drone is

35\text{ m}

North of the origin.

Adım Adım Çözüm

1

Determine initial facing direction and first displacement vector

The surveyor faces West at sunset. Moving

40\text{ m}

straight ahead places the drone at coordinates

(-40, 0)

relative to origin

(0,0)

. Current heading: West.

Sun sets in the West, establishing the initial cardinal reference.

2

Apply

135^\circ

clockwise turn and calculate second displacement

Turning

135^\circ

clockwise from West reorients the drone to North-East. Flying

30\sqrt{2}\text{ m}

adds

+30\text{ m}

East and

+30\text{ m}

North. Position becomes

(-40 + 30, 0 + 30) = (-10, 30)

.

A

135^\circ

clockwise shift from West (

270^\circ

) leads to North-East (

45^\circ

). Vector components are

30\sqrt{2}\cos(45^\circ) = 30

and

30\sqrt{2}\sin(45^\circ) = 30

.

3

Apply

90^\circ

right turn and calculate third displacement

Turning

90^\circ

right from North-East reorients heading to South-East. Flying

10\sqrt{2}\text{ m}

adds

+10\text{ m}

East and

-10\text{ m}

South. Position becomes

(-10 + 10, 30 - 10) = (0, 20)

.

A

90^\circ

right turn rotates North-East to South-East. Vector components are

+10\text{ m}

along East and

-10\text{ m}

along North.

4

Apply

135^\circ

counter-clockwise turn and calculate final displacement

Turning

135^\circ

counter-clockwise from South-East reorients heading to North. Flying

15\text{ m}

North advances position to

(0, 20 + 15) = (0, 35)

.

Turning

135^\circ

counter-clockwise from South-East points directly North.

5

Calculate net displacement magnitude and cardinal direction

Net coordinates are

(0, 35)

. Distance =

\sqrt{0^2 + 35^2} = 35\text{ m}

due North.

X-component is

0\text{ m}

and Y-component is

+35\text{ m}

.

Anahtar Kavram

Direction and Distance Test - Multi-step Angular Rotation & Vector Displacement

Soru 74Soru →

In a district administration department, a survey was conducted among $250$ officers regarding their technical expertise in three digital governance domains: Cyber Security ( $C$ ), Data Analytics ( $D$ ), and Public Grievance Portals ( $P$ ). The data collected is as follows:
- $120$ officers have expertise in Data Analytics.
- $115$ officers have expertise in Cyber Security.
- $100$ officers have expertise in Public Grievance Portals.
- $50$ officers have expertise in both Data Analytics and Cyber Security.
- $45$ officers have expertise in both Cyber Security and Public Grievance Portals.
- $40$ officers have expertise in both Data Analytics and Public Grievance Portals.
- $20$ officers possess expertise in all three digital governance domains.

Based on the given data, how many officers have expertise in EXACTLY TWO of these three domains?

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

The total number of officers who have expertise in exactly two digital governance domains is 75.

To find the number of officers with expertise in exactly two domains, we calculate the exclusive intersection regions by subtracting the central three-set intersection (

20

) from each of the given double-intersection totals:

(50 - 20) + (45 - 20) + (40 - 20) = 30 + 25 + 20 = 75

.

Adım Adım Çözüm

1

Calculate the number of officers in the region representing expertise in ONLY Data Analytics and Cyber Security

50 - 20 = 30

officers

The total intersection of Data Analytics and Cyber Security includes officers proficient in all three domains. Subtracting the 3-domain count isolates those proficient in only these two.

2

Calculate the number of officers in the region representing expertise in ONLY Cyber Security and Public Grievance Portals

45 - 20 = 25

officers

Subtracting the 3-domain count from the total intersection of Cyber Security and Public Grievance Portals isolates those proficient in only these two.

3

Calculate the number of officers in the region representing expertise in ONLY Data Analytics and Public Grievance Portals

40 - 20 = 20

officers

Subtracting the 3-domain count from the total intersection of Data Analytics and Public Grievance Portals isolates those proficient in only these two.

4

Sum the three mutually exclusive regions representing expertise in exactly two domains

30 + 25 + 20 = 75

officers

Adding these disjoint sets yields the total number of officers with expertise in exactly two of the three digital governance domains.

Anahtar Kavram

Principle of Inclusion-Exclusion and Venn Diagram Region Partitioning

Tahmini Süre:2m 0s

Soru 75Soru →

In a straight row of parked vehicles, a red car is positioned $14^{\text{th}}$ from the left end and $19^{\text{th}}$ from the right end. What is the total number of vehicles parked in the row?

32

33

31

34

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

The total number of vehicles parked in the row is 32.

When the rank of a single item is given from both ends of a row, the total number of items is calculated using

N = \text{Rank from Left} + \text{Rank from Right} - 1

. Adding

14

and

19

gives

33

, and subtracting

1

to correct for double-counting gives

32

.

Adım Adım Çözüm

1

Identify the given position values for the object.

Position from left (

L

) = 14, Position from right (

R

) = 19.

The rank of a single object is provided from both ends of a single linear row.

2

Apply the dual-rank single-person total formula.

Total (

N

) =

L + R - 1 = 14 + 19 - 1 = 32

.

The specified car is counted twice (once from the left and once from the right), so

1

must be subtracted from the sum of ranks.

Anahtar Kavram

Single-person dual rank total calculation

Soru 76Soru →

In a published civil services merit list, Rohan is ranked $14^{\text{th}}$ from the top, while Seema is ranked $33^{\text{rd}}$ from the bottom. After a re-evaluation of marks, Rohan and Seema interchange their positions, resulting in Rohan becoming $48^{\text{th}}$ from the top. A third candidate, Amit, holds a rank exactly halfway between Rohan's new position and Seema's new position. If $6$ candidates ranked above Amit withdraw their candidature and $4$ new candidates are placed below Amit in the updated list, what is Amit's final rank from the bottom of the modified merit list?

Cevabı ve açıklamayı göster

Cevap: 54

Cevap

Amit's final rank from the bottom of the modified merit list is 54.

Seema's dual information (

48^{\text{th}}

from top and

33^{\text{rd}}

from bottom) gives an initial total of 80 candidates. Amit's midpoint position between

14^{\text{th}}

and

48^{\text{th}}

is

31^{\text{st}}

from top, placing 49 candidates below him. Accounting for 6 withdrawals above and 4 additions below leaves 53 candidates below Amit, establishing his final rank as

54^{\text{th}}

from the bottom.

Adım Adım Çözüm

1

Find total candidates in the original merit list using the interchange data.

Total candidates N = 80

When Rohan and Seema interchange positions, Rohan occupies Seema's original spot (

48^{\text{th}}

from top). Since Seema was

33^{\text{rd}}

from bottom,

N = 48 + 33 - 1 = 80

.

2

Calculate Amit's initial rank from the top.

Amit's initial top rank = 31st

Amit is positioned halfway between Seema's new position (

14^{\text{th}}

from top) and Rohan's new position (

48^{\text{th}}

from top), so rank =

(14 + 48) / 2 = 31^{\text{st}}

.

3

Find initial counts of candidates ranked above and below Amit.

30 candidates above, 49 candidates below

With a top rank of

31^{\text{st}}

out of 80, there are

31 - 1 = 30

candidates above Amit and

80 - 31 = 49

candidates below Amit.

4

Adjust counts based on candidate withdrawals and additions.

24 candidates above, 53 candidates below

Subtract 6 from candidates above (

30 - 6 = 24

) and add 4 to candidates below (

49 + 4 = 53

).

5

Calculate Amit's revised rank from the bottom.

54th from the bottom

The rank from the bottom equals (candidates below + 1), which gives

53 + 1 = 54

.

Anahtar Kavram

Dual Rank Position Interchange, Midpoint Computation, and Dynamic List Adjustment

Tahmini Süre:2m 30s

Soru 77Soru →

In a district administration office, a survey of $120$ officers was conducted regarding their proficiency in using three government portals: e-Office ( $E$ ), CPGRAMS ( $C$ ), and GeM ( $G$ ). The survey revealed that $65$ officers are proficient in e-Office, $55$ in CPGRAMS, and $50$ in GeM. Furthermore, $25$ officers are proficient in both e-Office and CPGRAMS, $20$ in both CPGRAMS and GeM, and $22$ in both e-Office and GeM. If $10$ officers are proficient in all three portals, how many officers are proficient in none of these three portals?

7

17

27

37

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The number of officers proficient in none of the three portals is 7.

The total number of officers in at least one portal is calculated using

|E \cup C \cup G| = (65 + 55 + 50) - (25 + 20 + 22) + 10 = 113

. Subtracting this from the total group of

120

gives

120 - 113 = 7

officers in none of the portals.

Adım Adım Çözüm

1

Apply the Principle of Inclusion-Exclusion for three sets to find the total number of officers proficient in at least one portal.

|E \cup C \cup G| = |E| + |C| + |G| - |E \cap C| - |C \cap G| - |E \cap G| + |E \cap C \cap G|

Overlapping regions must be subtracted to prevent double counting, and the central triple intersection must be added back because it is subtracted three times.

2

Substitute the given numerical values into the formula.

|E \cup C \cup G| = 65 + 55 + 50 - 25 - 20 - 22 + 10 = 170 - 67 + 10 = 113

Calculating the total number of officers belonging to at least one category.

3

Subtract the number of officers proficient in at least one portal from the total officer population.

Number in none

= 120 - 113 = 7

Officers outside all three sets represent the complement of the union set.

Anahtar Kavram

Principle of Inclusion-Exclusion for 3 overlapping sets

Soru 78Soru →

If 1st January 2004 was a Thursday, what day of the week was 1st January 2005?

Cevabı ve açıklamayı göster

Cevap: Saturday; saturday

Cevap

Saturday

The correct answer is Saturday. The year 2004 is a leap year containing 366 days. Since the time period from 1st January 2004 to 1st January 2005 includes February 29th, there are 2 odd days. Advancing 2 days from Thursday results in Saturday.

Adım Adım Çözüm

1

Determine if the year 2004 is a leap year.

2004 is divisible by 4, so it is a leap year containing 366 days.

Leap years include an extra day (29th February).

2

Calculate the number of odd days between 1st January 2004 and 1st January 2005.

366 days divided by 7 equals 52 weeks and 2 odd days (

366 \pmod 7 = 2

).

Full weeks do not change the day of the week, so only the remainder (odd days) shifts the day.

3

Advance the given day by the number of odd days.

Thursday + 2 days = Saturday.

Adding 2 odd days to Thursday moves the day forward to Saturday.

Anahtar Kavram

Calculating day of the week shifts across leap years using odd days

Soru 79Soru →

In a state administrative audit of $300$ urban development projects, participation across three infrastructure domains—Sanitation ( $S$ ), Transport ( $T$ ), and Green Space ( $G$ )—was analyzed. The audit revealed that $145$ projects involve Sanitation, $135$ involve Transport, and $125$ involve Green Space. Exactly $70$ projects involve precisely two domains, while $20$ projects involve all three domains. Furthermore, $45$ projects involve both Sanitation and Transport, and $35$ projects involve both Transport and Green Space. Which of the following statements are correct? (Select all correct options.)

Geçerli olan tümünü seçin

Exactly

70

projects are dedicated exclusively to Sanitation.

The total number of projects involving at least two domains is

90

.

The number of projects involving both Green Space and Sanitation, but not Transport, is

30

.

Exactly

15

projects are not affiliated with any of the three domains.

Cevabı ve açıklamayı göster

Cevap: Exactly $70$ projects are dedicated exclusively to Sanitation.; The total number of projects involving at least two domains is $90$ .; The number of projects involving both Green Space and Sanitation, but not Transport, is $30$ .

Cevap

The statements asserting that exactly 70 projects are dedicated exclusively to Sanitation, that 90 projects involve at least two domains, and that 30 projects involve both Green Space and Sanitation but not Transport are all correct.

The three correct options accurately reflect the regional values obtained from set decomposition: exclusive Sanitation projects equal 70, projects in two or more domains total 90 (70 + 20), and projects in Green Space and Sanitation only total 30.

Adım Adım Çözüm

1

Calculate the regions involving exactly two domains

Sanitation and Transport only =

45 - 20 = 25

; Transport and Green Space only =

35 - 20 = 15

. Green Space and Sanitation only =

70 - (25 + 15) = 30

.

The intersection of two sets includes the triple intersection of all three sets.

2

Calculate single-domain exclusive regions

Sanitation only =

145 - (25 + 30 + 20) = 70

; Transport only =

135 - (25 + 15 + 20) = 75

; Green Space only =

125 - (30 + 15 + 20) = 60

.

Subtracting all overlapping two-set and three-set regions from each total set size isolates the single-domain counts.

3

Calculate projects in at least two domains

Projects in at least two domains = (Exactly 2 domains) + (All 3 domains) =

70 + 20 = 90

.

'At least two' is the union of the exactly two and exactly three regions.

4

Calculate total union and unaffiliated projects

Total in at least one domain =

70 + 75 + 60 + 25 + 15 + 30 + 20 = 295

. Unaffiliated projects =

300 - 295 = 5

.

Subtracting the union of all three sets from the total project pool yields the count of elements outside all sets.

Anahtar Kavram

Three-Set Venn Diagram Region Decomposition and Inclusion-Exclusion Principle

Soru 80Soru →

Five executives—Ford, Greta, Harry, Ian, and Julia—are seated in a single straight row facing South during a board meeting.

Based on the following conditions, arrange the executives in the exact order they are seated from West to East (i.e., from the viewer's left to right):

1. Ford sits at the extreme left end of the row.
2. Julia sits immediately to the right of Ford.
3. Greta sits second to the right of Ian.
4. Harry sits adjacent to Ian, but does not sit next to Julia.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct seating order from West to East (viewer's left to right) is Greta, Harry, Ian, Julia, and Ford.

By accounting for the South-facing orientation, we reverse the standard left and right. Ford at the 'extreme left' places him on the viewer's far right (East). Following the relative positioning places Julia beside him, Ian in the middle, Harry at position 2, and Greta at position 1 (West).

Adım Adım Çözüm

1

Determine the layout based on the facing direction.

Create five seats from West (viewer's left) to East (viewer's right). Since they face South, an executive's 'left' corresponds to East and their 'right' corresponds to West.

Crucial for correctly interpreting spatial clues without reversing the arrangement.

2

Place Ford and Julia using clues 1 and 2.

Ford is placed at position 5 (East/extreme left). Julia is placed at position 4 (immediate right of Ford).

Ford is at the extreme left (which is East when facing South). Julia is to his immediate right (which is West).

3

Place Ian and Greta using clue 3.

Ian is placed at position 3 and Greta is placed at position 1.

Greta is second to the right of Ian. The only remaining spaces with a gap of one seat are positions 1 and 3 (right = West).

4

Place Harry using clue 4.

Harry is placed at position 2.

Harry must be adjacent to Ian (position 3), which means he must be at position 2 or 4. Position 4 is occupied by Julia, so Harry takes position 2. This perfectly satisfies the condition that he is not next to Julia.

Anahtar Kavram

Linear Seating Arrangement with South-Facing Orientation

General Mental Ability

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram